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Identify the matching numerator and denominator
Covariance has the product of X and Y units; the denominator has the same product. The ratio is dimensionless when both standard deviations are positive and finite.
Check the covariance bound on the stated scale
For finite variances, the magnitude of covariance cannot exceed the product of the standard deviations. Dividing by that positive product therefore gives a correlation between −1 and 1. Covariance itself is not universally bounded by ±1 because its scale depends on the variables.
Do not divide by a zero standard deviation
If one variable is constant, its standard deviation is zero. The usual Pearson-correlation expression is then undefined, not automatically zero. A zero covariance numerator does not repair a zero denominator.
Distinguish linear association from independence
Correlation summarizes standardized linear co-movement. A zero correlation does not generally establish independence. A non-linear relationship can remain even when positive and negative covariance contributions cancel.